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We describe a tracking algorithm to address the interactions among objects, and to track them individually and confidently via a static camera. It is achieved by constructing an invariant bipartite graph to model the dynamics of the tracking process, of which the nodes are classified into objects and profiles. The best match of the graph corresponds to an optimal assignment for resolving the identities of the detected objects. Since objects may enter/exit the scene indefinitely, or when interactions occur/conclude they could form/leave a group, the number of nodes in the graph changes dynamically. Therefore it is critical to maintain an invariant property to assure that the numbers of nodes of both types are kept the same so that the matching problem is manageable. In addition, several important issues are also discussed, including reducing the effect of shadows, extracting objects' shapes, and adapting large abrupt changes in the scene background. Finally, experimental results are provided to illustrate the efficiency of our approach.